Theorem alrimii 31819

Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
alrimii.1	|- F/ y ph
alrimii.2	|- ( ph -> ps )
alrimii.3	|- ( [. y / x ]. ch <-> ps )
alrimii.4	|- F/ y ch

Assertion

Ref	Expression
alrimii	|- ( ph -> A. x ch )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, y)

Proof of Theorem alrimii

Step	Hyp	Ref	Expression
1		alrimii.1	. . 3 |- F/ y ph
2		alrimii.2	. . . 4 |- ( ph -> ps )
3		alrimii.3	. . . 4 |- ( [. y / x ]. ch <-> ps )
4	2, 3	sylibr 214	. . 3 |- ( ph -> [. y / x ]. ch )
5	1, 4	alrimi 1903	. 2 |- ( ph -> A. y [. y / x ]. ch )
6		nfsbc1v 3299	. . 3 |- F/ x [. y / x ]. ch
7		alrimii.4	. . 3 |- F/ y ch
8		sbceq2a 3291	. . 3 |- ( y = x -> ( [. y / x ]. ch <-> ch ) )
9	6, 7, 8	cbval 2050	. 2 |- ( A. y [. y / x ]. ch <-> A. x ch )
10	5, 9	sylib 198	1 |- ( ph -> A. x ch )

Syntax hints:   -> wi 4   <-> wb 186   A.wal 1405   F/wnf 1639   [.wsbc 3279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382

This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-sbc 3280

This theorem is referenced by: (None)